Linear Algebra, Fall 2007 Quiz 4 Hints 1. Let A be a 6 × 5 matrix. If a 1 , a 2 , a 4 are linearly independent and a 3 = a 1 + a 2 , a 5 = a 1-a 2 + 2 a 4 , ﬁnd the reduced row echelon form of A and a basis of the column space of A . The proof of Theorem 3.6.5 (and Example 3) explains how to ﬁnd a basis for the row space of A and the null space of A . The paragraph before Theorem 3.6.6, the proof of Theorem 3.6.6 and Example 4 explains how to ﬁnd a basis for the column space of A . Here the answer is         1 0 1 0 1 0 1 1 0-1 0 0 0 1 2 0 0 0 00 0 0 0 00 0 0 0 00         . In short, a 1 , a 2 , a 4 correspond to lead variables. a 3 , a 5 correspond to free variables. It is also clear that a 1 , a 2 , a 4 span and form a basis of the column space. If you don’t know why, read the above materials in detail. 2. Give an example of a (non-trivial) linear transformation from C [0 , 1] to R . Explain. Some examples are given in section 4.1, Problem 10 and parts of Problem 11. 3. True or False? Explain. Let L be a linear transformation from

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